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However, this will turn ALL terms checkboxes to radio buttons.

Not only that, it'll turn any checkbox in a meta box - not ideal!

Instead, let's specifically target the wp_terms_checklist() function, which is used to generate the list of checkboxes across the admin (including quick edit).

/**
 * Use radio inputs instead of checkboxes for term checklists in specified taxonomies.
 *
 * @param   array   $args
 * @return  array
 */
function wpse_139269_term_radio_checklist( $args ) {
    if ( ! empty( $args['taxonomy'] ) && $args['taxonomy'] === 'category' /* <== Change to your required taxonomy */ ) {
        if ( empty( $args['walker'] ) || is_a( $args['walker'], 'Walker' ) ) { // Don't override 3rd party walkers.
            if ( ! class_exists( 'WPSE_139269_Walker_Category_Radio_Checklist' ) ) {
                /**
                 * Custom walker for switching checkbox inputs to radio.
                 *
                 * @see Walker_Category_Checklist
                 */
                class WPSE_139269_Walker_Category_Radio_Checklist extends Walker_Category_Checklist {
                    function walk( $elements, $max_depth, $args = array() ) {
                        $output = parent::walk( $elements, $max_depth, $args );
                        $output = str_replace(
                            array( 'type="checkbox"', "type='checkbox'" ),
                            array( 'type="radio"', "type='radio'" ),
                            $output
                        );
    
                        return $output;
                    }
                }
            }
    
            $args['walker'] = new WPSE_139269_Walker_Category_Radio_Checklist;
        }
    }

    return $args;
}

add_filter( 'wp_terms_checklist_args', 'wpse_139269_term_radio_checklist' );

We hook onto the wp_terms_checklist_args filter, then implement our own custom "walker" (a family of classes used to generate hierarchical lists). From there, it's a simply string replace of type="checkbox" with type="radio" if the taxonomy is whatever we've configured it to match (in this case "category").